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arXiv · 1508.06815

A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory

Yuki Arano·Yosuke Kubota
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Abstract

We investigate the homological ideal $\mathfrak{J}_G^H$, the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah-Segal completion theorem with the comparison of $\mathfrak{J}_G^H$ with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant $\mathrm{KK}$-theory, McClure's restriction map theorem, permanence property of the Baum-Connes conjecture under extensions of groups and a class of $\mathfrak{J}_G$-injective objects coming from $\mathrm{C}^*$-dynamical systems, continuous Rokhlin property.

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Yuki Arano, Yosuke Kubota. 2015-12-22. A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory. https://arxiv.org/abs/1508.06815

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